Abstract

Dynamic scaling behavior has been observed during the room-temperature growth of sputtered
Au films on SiO2using the atomic force microscopy technique. By the analyses of the dependence of
the roughness, σ, of the surface roughness power,P(f), and of the correlation length,ξ, on the film thickness,h, the roughness exponent,α = 0.9 ± 0.1, the growth exponent,β = 0.3 ± 0.1, and the dynamic scaling exponent,z = 3.0 ± 0.1 were independently obtained. These values suggest that the sputtering
deposition of Au on SiO2at room temperature belongs to a conservative growth process in which the Au grain
boundary diffusion plays a dominant role.

Keywords:

Introduction

Thin films having 0.1 nm thickness play important roles in various fields of modern
day science and technology [1,2]. In particular, the structure and properties of metal films on non-metal surfaces
are of considerable interest [3-6] due to their potential applications in various electronic, magnetic, and optical
devices. Most of these properties change drastically, when ultrathin films are formed
from bulk materials, because of the confinement effects. The study of the morphology
of thin films with the variation of thickness gives an idea about the growth mechanism
of these films [7,8]. This indicates the importance of such studies both from basic theoretical understanding
and applications points of view. The study of morphology and the understanding of
growth mechanisms are also essential to prepare materials in controlled way for the
desired properties. Scanning probe microscopy techniques, such as atomic force microscopy
(AFM), are important methodologies to study the surface morphology in real space [9-12]. The top surface can be imaged using an AFM and these images provide information
about the morphology and the variation of roughness as a function of thickness and
scan length. This variation of roughness essentially gives the height–height correlation
and can be used to extract the growth mechanism of the film [13].

All rough surfaces exhibit perpendicular fluctuations which are characterized by a
rms width being with h(xy) the height function and <….> the spatial average over a planar reference surface.
Films grown under nonequilibrium condition are expected to develop self-affine surfaces
[7,14], whose rms widths scale with time t and the length L sampled as [15]

(1)

where for and for . The parameter 0 < α < 1 is defined as the roughness exponent [16], and the parameter, β, as the growth exponent. Actual self-affine surfaces are characterized by an upper
horizontal cutoff to scaling, or correlation length, ξ, beyond which the surface width no longer scales as Lα, and eventually reaches a saturation value, σ. Implicit in Eq. 1 is a correlation length which increases with time as , where z = α/β is the dynamic scaling exponent.

In thin films deposition methodologies in which the film thickness,h, is proportional to the time of deposition,t, then, in the asymptotical limits,

(2)

(3)

whereaandbare the opportune proportionality constants.

Theoretical treatments of nonequilibrium film growth typically employ partial differential
equations involving phenomenological expansions in the derivatives of a time-dependent
height function, h(xyt). The Kardar–Parisi–Zhang (KPZ) equation [17] and the Siegert–Plischke (SP) equation [8] are examples of this approach. The KPZ equation concerns the nonconservative systems
(it does not conserve the particle number): in the nonconservative dynamics the side
growth is allowed with the creation of voids and overhangs, but the relaxation mechanisms
such as desorption or diffusion are not dominant enough to eliminate these defects
completely. The KPZ equation for nonequilibrium and nonconservative systems yields
α = 0.3–0.4 and β = 0.24–0.25 for growth of a two-dimensional surface [18,19]. The SP equation concerns, instead, nonequilibrium but conservative systems. For
conservative growth [8,20-23] the primary relaxation mechanism is the surface diffusion. Because the desorption
of atoms and formation of overhangs and voids are negligibly small, the mass and volume
conservation laws play an important role in the growth. The SP equation for nonequilibrium
and conservative systems yields α = 1 and β = 0.25 for growth of a two-dimensional surface [8]. The values of α and β predicted by the theories for nonconservatives and conservatives systems may vary
depending on the couplings with other effects.

Although extensive theoretical studies have predicted many important features in the
growth dynamics of thin films, experimental works have to be performed to verify these
predictions. In this article, we report an AFM study of the thickness dependence of
σ and ξ for a nanostructured thin Au film deposited by sputtering at room temperature on
a SiO2 substrate. By such, studies the value of α = 0.9 ± 0.1 and β = 0.3 ± 0.1 are determined. Independently, the value of 1/z = 0.3 ± 0.1 is obtained. From these measured values, we suggest that the growth of
Au film on SiO2 at room temperature is consistent with a conservative growth process. A comparison
with theoretical and experimental literature data on the growth of thin metal films
is finally performed. The Au/SiO2 system has been chosen for two primary reasons: (1) the Au/SiO2 interface grows, at room temperature, in the Volmer–Weber mode, and it is unreactive
and abrupt [24]. This fact simplifies the experimental analyses allowing to neglect spurious effects
on the interface growth deriving from the reaction between the deposited film and
the substrate. From this fact, after all, follows that the growth of Au film on SiO2 at room temperature belongs to the conservative class of dynamic process; (2) The
Au/SiO2 nanostructured system represents a widely investigated material for nanoelectronic
applications [25]—in such a system, the reaching of an atomic level control of the structural properties
allow a manipulation of the nanoscale electrical ones [25].

Experimental

A cz-<100> silicon wafer (with resistivity, ) was used as starting substrate. It was initially etched in 10% aqueous HF solution
to remove the native oxide. Then it was annealed at 1223 K for 15 min in O2in order to grow an uniform, 10-nm thick, amorphous SiO2layer. A series of Au films were deposited onto the SiO2substrate by RF sputtering using an Emitech K550× Sputter coater apparatus. The depositions
were performed at room temperature, with a base pressure of 10−4 Pa. Samples of increasing nominal Au thickness,h, were deposited: 2 nm (sample 1), 8 nm (sample 2), 14 nm (sample 3), 20 nm (sample
4), 26 nm (sample 5), 32 nm (sample 6). In our experimental deposition conditions,
the thickness,h, of the deposited Au film is proportional to the deposition timet:h = at being . The nominal thickness of the deposited Au film was checked by Rutherford backscattering
analyses (using 2 MeV4He+backscattered ions at 165°). The evolution of Au film morphology with the thickness,h, was analyzed by AFM using a PSIA XE150 microscope operating in non-contact mode
and ultra-sharpened Si tips were used and substituted as soon as a resolution loss
was observed during the acquisition. AFM images were analyzed by using the XEI software.
The XEI is the PSIA-AFM image processing and analysis program. The XEI software allows
users to extract several information from the sample surface by utilizing various
analysis tools and also by providing the ability to remove certain artifacts from
scan data. For example, its analysis functions include to profile tracer and region,
line measurement of height, line profile, power spectrum, line histogram, regional
measurement of height, average roughness, volume, surface area, histogram, bearing
ratio, and grain analysis functions.

Results and Discussion

The change in morphology of the Au film as a function of its thickness, h, has been followed by AFM. From such analyses, the Au film, in all the samples, results
to be formed by spherical nanometric grains of increasing mean size [26]. As an example, Fig. 1 shows 5 × 5 μm AFM representative images of the samples: (a) the starting SiO2 substrate, (b) sample 1 (h = 2 nm), (c) sample 2 (h = 8 nm), (d) sample 3 (h = 14 nm), (e) sample 4 (h = 20 nm), (f) sample 5 (h = 26 nm), (g) sample 6 (h = 32 nm), respectively. First, we obtained the roughness σ for each sample by the corresponding AFM images using the XEI software. In particular,
the value of σ for each sample was calculated by averaging the values obtained by five 5 × 5 μm
AFM images (for which the roughness results saturated with the scan size L). The error in σ was deducted by the averaging procedure. Thus, Fig. 2 reports the values of σ obtained as a function of h: the experimental data (dots) were fitted by Eq. 2 (continuous line) obtaining the
growth exponent β = 0.3 ± 0.1.

Figure 2. Experimental (dots) values of the saturated surface roughness of the Au film as a
function of the film thickness and fit (continuous line) by Eq. 2. The fit parameter
β resultedβ = 0.3 ± 0.1

Furthermore, for each sample we calculated also the averaged power spectrum from the
spectra of each of the 512 linear traces. Thus, in contrast to σ, the power spectra are calculated from one-dimensional cross sections of the surface.
Each spectrum is the square of the surface roughness amplitude per spatial frequency
interval and the integral over all frequencies is the mean-square surface roughness
within the measured bandwidth (σ2). Thus, Fig. 3 reports the calculated surface roughness power, P, as a function of the frequency, f, concerning the representative AFM images presented in Fig. 1: Figure 3a for the sample 1 (h = 2 nm), Fig. 3b for the sample 2 (h = 8 nm), Fig. 3c for the sample 3 (h = 14 nm), Fig. 3d for the sample 4 (h = 20 nm), Fig. 3e for the sample 5 (h = 26 nm), and Fig. 3f for the sample 6 (h = 32 nm), respectively. The power spectra in Fig. 3 have two distinct regions. The flat, low frequency part resembles uncorrelated white
noise. The sloped portion represents the correlated portion of the surface roughness.
To obtain the roughness exponent α from this data, we fit the power law decay (in the linear region in the log–log plot)
to

Figure 3. Representative surface roughness power spectra for the analyzed sample calculated
by the AFM images reported in Fig. 1:afor the sample with a thickness of 2 nmbof 8 nm,cof 14 nm,dof 20 nm,eof 26 nm,fof 32 nm of Au respectively. The continuous lines represent the fit by Eq. 4. The
values ofγireported as insets are calculated by such fits

By the values of β = 0.3 ± 0.1 and α = 0.9 ± 0.1 previously derived, the value of the dynamic scaling exponent z = α/β = 2.9 ± 0.4 (or alternatively of 1/z = 0.3 ± 0.1) is predicted. But now -z can be derived by the experimental data to try confirmation of the theoretical predicted
value. In fact, to characterize the scale of correlations perpendicular to the growing
direction, the correlation frequency can be used. It can be evaluated by the power spectra as the spatial frequency where
P(f) has fallen to 1/e of its saturation low frequency value and above which σ is correlated. Using the five power spectra for each sample already used for the
calculation of the γi and performing the averaging procedure, the values of for the correlation lengths for the samples 1, 2, 3, 4, 5, and 6, respectively, were
obtained. Figure 4 reports as dots, in a log–log scale, such values as a function of the film thickness,
h. The continuous line is the fit by Eq. 3 allowing the determination of 1/z = 0.3 ± 0.1 in agreement with the predicted value. Finally, from the AFM analyses
reported in Fig. 1, statistical data on the radius, area and volume of the Au nanometric grains forming
the film can be obtained. The XEI software for the analyses of the AFM images allow
to obtain the distribution of the grains radii, R, and of the grains areas S by a procedure consisting in the definition of each grain area by the surface image
sectioning of a plane that was positioned at the half grain height. As a consequence,
the distribution of the grains radii R, surface areas, S, and volumes, V can be extracted. By such distributions, the mean grain radius, <R>, the mean grain area, <S>, and the mean grain volume <V> can be extracted with the respective statistical errors. Therefore, Fig. 5a–c report <R>, <S>, and <V> as a function of the film thickness h. As a final remark, it is worth to note that Fig. 5c, being indicates clearly a grain growth scaling law . Since the dynamical scaling theories predict [8] then also such a data conduct to the results z = 3 for the dynamic scaling exponent.

Figure 4. Experimental (dots) values of the correlation length for the Au film as a function
of the film thickness and fit (continuous line) by Eq. 3. The fit parameter 1/zresulted 1/z = 0.3 ± 0.1

Figure 5. Experimental evolution (dots) of the mean grain radius <R> (a), mean grain surface area <S> (b), and mean grain volume <V> (c) as a function of the thickness filmh. The lines are only guide for the eyes

Now, we turn to the comparison of the data presented in this study with experimental
and theoretical literature studies. The values obtained by us in this study are comparable
to those reported by Chevrier et al. [28] (β = 0.25–0.32) for vapor-deposited Fe on Si at 323 K, by G. Palasantzas and J. Krim
[29] (α = 0.82 ± 0.05, β = 0.29 ± 0.06 and z = 2.5 ± 0.5) for room-temperature vapor-deposited Ag film on quartz. But they do
not coincide with the values reported by You et al. [30] (α = 0.42, β = 0.40) for room-temperature sputtered Au film on Si, to those reported by Fanfoni
et al. [31] and Placidi et al. [32] for the molecular beam epitaxy dynamical growth of silver islands on GaAs(001)-(2
× 4) (z = 1.5 ± 0.2 and z = 4.2 ± 0.4, respectively) and to those reported by Rosei et al. [33] for reactive-deposited Ge on Si(1111) (z = 0.70 ± 0.20). We can attribute the difference of our results from those of You
et al. to the different used substrates used since though Au is unreactive with SiO2, it is reactive with Si [34] and to the lower substrate temperature. The difference with respect to the values
of Fanfoni et al., Placidi et al. and Rosei et al. can be attributed to differences
in film deposition conditions. We believe that our values of α = 0.9 ± 0.1, β = 0.3 ± 0.1 and 1/z = 0.3 ± 0.1 for room-temperature sputtered Au films are more consistent with a conservative
deposition process (i.e. prediction of the SP equation) rather than a nonconservative
one (i.e. prediction of the KPZ equation). Other experiments that characterize self-affine
fractals using different techniques [35-37] indicate that the values of α measured from metal thin films range from 0.65 to 0.95, which are indeed higher than
that predicted by the nonconservative growth models [17-19]. The exponents obtained in this experiment are thus more consistent with the results
of conservative growth models [20-23]. A justification of this fact can be found in the microscopic mechanism governing
the Au film growth on SiO2 at room temperature. Our recent data [26] suggest that during the Au sputter deposition at room temperature the film growth
is driven by the Au grain boundary diffusion with a diffusion coefficient (rather than an Au surface diffusion, since the surface diffusion coefficient of
Au on SiO2 is very small at room temperature, [24]). In fact, the AFM analyses in connection with transmission electron microscopy analyses
allow to conclude that the Au film is formed by three-dimensional nanometric grains
that grows as “normal grains” for thickness in the 0.33 nm. For higher thickness,
together with the normal grain growth, the growth of “abnormal large grains” is observed.
The normal grain growth appears to be (at room temperature) controlled by Au diffusion
on grain boundaries (rather than by Au surface diffusion) while the abnormal grain
growth process appears to be driven by the differences between surface energies of
the normal and abnormal grains, so that grains with favored orientations grow at a
higher rate (with respect to the normal grain growth rate) by annihilating the surrounding
normal grains. We believe, thus, that, during the deposition process, the overhangs
and voids are unlikely to appear in the growth of the film because the Au grain boundary
diffusion plays a dominant role.

Conclusion

An AFM study of the dynamic evolution of a growing interface was carried out for room-temperature
Au sputtered onto a SiO2 substrate. The analyses of AFM images of the Au film allowed us to derive the roughness,
σ, the surface roughness power, P(f), and the correlation length, ξ, as a function of the film thickness, h. Analyzing such dependences the roughness exponent, the growth exponent and the dynamic
scaling exponent were independently obtained: α = 0.9 ± 0.1, β = 0.3 ± 0.1 and z = 3.0 ± 0.1. These values suggest that the sputtering deposition of Au on SiO2 at room temperature belongs to a conservative growth process in which the Au grain
boundary diffusion plays a dominant role. This study suggests further analyses concerning,
for example, the dependence of the exponents αβ, and z on the substrate temperature during the film deposition (such as pointed out in the
experimental study of You et al. [30] for the case of Au on Si), on the rate deposition (such as pointed out by Collins
et al. [38]) and the extension of the experimental investigation to other systems that could
present nonequilibrium conservative or nonconservative dynamical growth mechanisms
(e.g., Pd/SiO2, Au/SiC, Pd/SiC, Au/GaN, Pd/GaN, Pd/Si).